If $9 + f''(x) + f'(x) = x^2 + f^2(x)$, where $f(x)$ is twice differentiable function such that $f''(x) \neq 0 \forall x \in R$ and let $P$ be the point of maxima of $f(x)$ then find the number of tangents which can be drawn from $P$ to the circle $x^2 + y^2 = 9$.
Step-by-Step Solution
Key Concept: At a maximum of $f(x)$, the second derivative condition combined with the given differential equation constrains the point to lie outside a specific circle, precluding tangent existence.
Given $9 + f''(x) + f'(x) = x^2 + f^2(x)$, or equivalently $9 + \frac{d^2y}{dx^2} + \frac{dy}{dx} = x^2 + y^2$. At a point of maxima of $f(x)$, we have $\frac{dy}{dx} = 0$ and $\frac{d^2y}{dx^2} 9$. Therefore $P(x,y)$ satisfies $x^2 + y^2 > 9$, placing it outside the circle $x^2 + y^2 = 9$. Since the locus of points where tangent lines to curves would originate must lie on or outside this circle, no tangent to any curve can exist at these maxima points.
Correct Answer: 0